This type of things is right up my alley, so I decided to have a look. First of all, here is a collection of some data tabulated from the CPI report:

The published percentage change in each constituent were only given to the nearest 10 bps, so I decided to grab the actual seasonally adjusted index values and compute the percentage changes myself for more accuracy. The numbers agree well enough, so there is nothing fishy here so far.

In fact, if you take the three percentage changes for Shelter, Fuels and utilities, and Household furnishings and operatiors (highlighted in yellow) and compute the weighted sum, it agrees precisely with the percentage change in the Housing number (as it should). Explicitly,

which is precisely the number you get by looking at the percentage change in the adjusted Housing index.

So far so good…

Now, if we dig a little deeper into Fuels and utility, we find

In other words, the weighted sum of the constituents agrees with the percentage change of the adjusted Fuels and utilities index to within 2 bps. I might expect closer agreement, but there is still nothing too shocking.

Things get weird when we look at the Shelter numbers

Here we see again that the published numbers agree well with the percentage changes in the adjusted indexes, but there is something odd. If we compute the weighted sum of the constituents as we did for Fuel and utilities and Housing, we get -0.10% as opposed to -0.48% as we’d expect. This is a 38 bps different that could change a negative CPI number into a positive one.

What is going on?

There are two primary possible sources of error: 1.) the weights are wrong, 2.) the index values are wrong. I do not list percentage change of each constituent as a possible source of error because I verified these against the index values provided.

Of course, the third (and not unlikely) possible source of error is that I am misinterpreting the data. If anyone has a good explanation for this apparent discrepancy, please let me know.